Problem: The sum of the first $20$ positive even integers is also the sum of four consecutive even integers. What is the largest of these four integers?
Explanation: The sum of the first 20 positive even integers is $2 + 4 + \dots + 40 = 2 (1 + 2 + \dots + 20)$.  For all $n$, $1 + 2 + \dots + n = n(n + 1)/2$, so $2 (1 + 2 + \dots + 20) = 20 \cdot 21 = 420$.

Let the four consecutive even integers be $n - 6$, $n - 4$, $n - 2$, and $n$.  Their sum is $4n - 12 = 420$, so $n = \boxed{108}$.